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Cayley diagram

C7×GO+(4,𝔽2) #92741

order504
isoclinismisoclinic class of GO+(4,𝔽2)
automorphismAut(G) = C2×C3×AΓL(1,𝔽9) (order 864) · Inn(G) = GO+(4,𝔽2) · Out(G) = C22×C3
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×GO+(4,𝔽2)
unambiguous semidirect product#157⋊(C2×C7)
unambiguous semidirect productD32⋊(C2×C7)

Splittings

Fitting, socle, HallC32×C7 ⋊ D4
SylowC32 ⋊ C7×D4

Subgroups

subgroupG/H
center, hypercenterC7GO+(4,𝔽2)
commutator, cosocleC3⋊D3C22×C7
FrattiniC1G
Fitting, socleC32×C7D4
hypocenterC32C7×D4
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C7

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 42

A minimal faithful permutation representation — 2 generators on 13 points:

(1,3)(2,5,6)(7,8,9,10,11,12,13);(1,2)(3,6)(4,5)

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC7×AΓL(1,𝔽9), #11759722, D7×GO+(4,𝔽2), …
maximal normal subgroup ofC7×AΓL(1,𝔽9), #11759722, D7×GO+(4,𝔽2), …
maximal quotient ofC7×D32⋊C4, C7×#3773, C7×#3774, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 167);